A geometric discretisation scheme applied to the Abelian Chern-Simons theory

TL;DR

This paper presents a geometric discretisation scheme for Abelian Chern-Simons theory that accurately captures topological features and yields triangulation-independent partition functions matching the continuum theory.

Contribution

It introduces a novel geometric discretisation approach for Abelian Chern-Simons theory, demonstrating triangulation independence and explicit computation of the partition function.

Findings

Discretisation scheme captures topological features accurately

Partition function is triangulation independent

Results match continuum partition function

Abstract

We give a detailed general description of a recent geometrical discretisation scheme and illustrate, by explicit numerical calculation, the scheme's ability to capture topological features. The scheme is applied to the Abelian Chern-Simons theory and leads, after a necessary field doubling, to an expression for the discrete partition function in terms of untwisted Reidemeister torsion and of various triangulation dependent factors. The discrete partition function is evaluated computationally for various triangulations of $S^3$ and of lens spaces. The results confirm that the discretisation scheme is triangulation independent and coincides with the continuum partition function